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=Steady-State Tensor Virial Equations=
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=Steady-State 2<sup>nd</sup>-Order Tensor Virial Equations=
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By satisfying all nine components of the Second-Order Tensor Virial Equation, the entire set of Riemann S-Type Ellipsoids can be determined.
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By satisfying all six &#8212; not necessarily unique &#8212; components of the Second-Order Tensor Virial Equation, the entire set of Riemann Ellipsoids can be determined.
{{LSU_HBook_header}}
{{LSU_HBook_header}}
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==Equilibrium Expressions==
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[<b>[[User:Tohline/Appendix/References#EFE|<font color="red">EFE</font>]]</b> &sect;11(b), p. 22] <font color="#007700">Under conditions of a stationary state, [the tensor virial equation] gives,</font>
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<div align="center">
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<table border="0" cellpadding="5" align="center">
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<tr>
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  <td align="right">
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<math>~2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij} </math>
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  </td>
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  <td align="center">
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<math>~=</math>
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  </td>
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  <td align="left">
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<math>~- \delta_{ij}\Pi \, .</math>
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  </td>
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</tr>
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</table>
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</div>
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<font color="#007700">[This] provides six integral relations which must obtain whenever the conditions are stationary</font>.
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When viewing the (generally ellipsoidal) configuration from a rotating frame of reference, the 2<sup>nd</sup>-order TVE takes on the more general form:
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<table border="0" cellpadding="5" align="center">
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<tr>
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  <td align="right">
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<math>~0</math>
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  </td>
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  <td align="center">
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<math>~=</math>
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  </td>
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  <td align="left">
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<math>~
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2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij} + \delta_{ij}\Pi
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+ \Omega^2 I_{ij} - \Omega_i\Omega_k I_{kj} + 2\epsilon_{ilm}\Omega_m \int_V \rho u_lx_j dx
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\, .
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</math>
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  </td>
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</tr>
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<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 2, &sect;12, Eq. (64)</font> ]</td></tr>
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</table>
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EFE (p. 57) also shows that &hellip; <font color="#007700">The potential energy tensor &hellip; for a homogeneous ellipsoid is given by</font>
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<table border="0" cellpadding="5" align="center">
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<tr>
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  <td align="right">
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<math>~\frac{\mathfrak{W}_{ij}}{\pi G\rho}</math>
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  </td>
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  <td align="center">
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<math>~=</math>
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  </td>
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  <td align="left">
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<math>~-2A_i I_{ij} \, ,</math>
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  </td>
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</tr>
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<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 3, &sect;22, Eq. (128)</font> ]</td></tr>
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</table>
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<font color="#007700">where</font>
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<table border="0" cellpadding="5" align="center">
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<tr>
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  <td align="right">
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<math>~I_{ij}</math>
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  </td>
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  <td align="center">
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<math>~=</math>
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  </td>
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  <td align="left">
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<math>~\tfrac{1}{5} Ma_i^2 \delta_{ij} \, ,</math>
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  </td>
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</tr>
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<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 3, &sect;22, Eq. (129)</font> ]</td></tr>
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</table>
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<font color="#007700">is the moment of inertia tensor.</font>
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==Adopted (Internal) Velocity Field==
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EFE (p. 130) states that &hellip; <font color="#007700">The kinematical requirement, that the motion <math>~(\vec{u})</math>, associated with <math>~\vec{\zeta}</math>, preserves the ellipsoidal boundary, leads to the following expressions for its components:</font>
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<table border="0" cellpadding="5" align="center">
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<tr>
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  <td align="right">
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<math>~u_1</math>
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  </td>
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  <td align="center">
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<math>~=</math>
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  </td>
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  <td align="left">
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<math>~- \biggl[ \frac{a_1^2}{a_1^2 + a_2^2}\biggr] \zeta_3 x_2 + \biggl[ \frac{a_1^2}{a_1^2+a_3^2}\biggr] \zeta_2 x_3 \, ,</math>
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  </td>
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</tr>
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 +
<tr>
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  <td align="right">
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<math>~u_2</math>
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  </td>
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  <td align="center">
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<math>~=</math>
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  </td>
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  <td align="left">
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<math>~- \biggl[ \frac{a_2^2}{a_2^2 + a_3^2}\biggr] \zeta_1 x_3 + \biggl[ \frac{a_2^2}{a_2^2+a_1^2}\biggr] \zeta_3 x_1 \, ,</math>
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  </td>
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</tr>
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 +
<tr>
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  <td align="right">
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<math>~u_3</math>
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  </td>
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  <td align="center">
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<math>~=</math>
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  </td>
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  <td align="left">
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<math>~- \biggl[ \frac{a_3^2}{a_3^2 + a_1^2}\biggr] \zeta_2 x_1 + \biggl[ \frac{a_3^2}{a_3^2+a_2^2}\biggr] \zeta_1 x_2 \, .</math>
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  </td>
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</tr>
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<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;47, Eq. (1)</font> ]</td></tr>
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</table>
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==General Coefficient Expressions==
==General Coefficient Expressions==

Revision as of 21:18, 3 August 2020


Contents

Steady-State 2nd-Order Tensor Virial Equations

By satisfying all six — not necessarily unique — components of the Second-Order Tensor Virial Equation, the entire set of Riemann Ellipsoids can be determined.

Whitworth's (1981) Isothermal Free-Energy Surface
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Equilibrium Expressions

[EFE §11(b), p. 22] Under conditions of a stationary state, [the tensor virial equation] gives,

~2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij}

~=

~- \delta_{ij}\Pi \, .

[This] provides six integral relations which must obtain whenever the conditions are stationary.

When viewing the (generally ellipsoidal) configuration from a rotating frame of reference, the 2nd-order TVE takes on the more general form:

~0

~=

~
2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij} + \delta_{ij}\Pi 
+ \Omega^2 I_{ij} - \Omega_i\Omega_k I_{kj} + 2\epsilon_{ilm}\Omega_m \int_V \rho u_lx_j dx
\, .

[ EFE, Chapter 2, §12, Eq. (64) ]

EFE (p. 57) also shows that … The potential energy tensor … for a homogeneous ellipsoid is given by

~\frac{\mathfrak{W}_{ij}}{\pi G\rho}

~=

~-2A_i I_{ij} \, ,

[ EFE, Chapter 3, §22, Eq. (128) ]

where

~I_{ij}

~=

~\tfrac{1}{5} Ma_i^2 \delta_{ij} \, ,

[ EFE, Chapter 3, §22, Eq. (129) ]

is the moment of inertia tensor.

Adopted (Internal) Velocity Field

EFE (p. 130) states that … The kinematical requirement, that the motion ~(\vec{u}), associated with ~\vec{\zeta}, preserves the ellipsoidal boundary, leads to the following expressions for its components:

~u_1

~=

~- \biggl[ \frac{a_1^2}{a_1^2 + a_2^2}\biggr] \zeta_3 x_2 + \biggl[ \frac{a_1^2}{a_1^2+a_3^2}\biggr] \zeta_2 x_3 \, ,

~u_2

~=

~- \biggl[ \frac{a_2^2}{a_2^2 + a_3^2}\biggr] \zeta_1 x_3 + \biggl[ \frac{a_2^2}{a_2^2+a_1^2}\biggr] \zeta_3 x_1 \, ,

~u_3

~=

~- \biggl[ \frac{a_3^2}{a_3^2 + a_1^2}\biggr] \zeta_2 x_1 + \biggl[ \frac{a_3^2}{a_3^2+a_2^2}\biggr] \zeta_1 x_2 \, .

[ EFE, Chapter 7, §47, Eq. (1) ]


General Coefficient Expressions

As has been detailed in an accompanying chapter, the gravitational potential anywhere inside or on the surface, ~(a_1,a_2,a_3) ~\leftrightarrow~(a,b,c), of an homogeneous ellipsoid may be given analytically in terms of the following three coefficient expressions:


~A_1


~=

~2\biggl(\frac{b}{a}\biggr)\biggl(\frac{c}{a}\biggr)
\biggl[  \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, ,


~A_3


~=


~2\biggl(\frac{b}{a}\biggr) \biggl[  \frac{(b/a) \sin\theta - (c/a)E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr] \, ,


~A_2


~=

~2 - (A_1+A_3) \, ,

where, ~F(\theta,k) and ~E(\theta,k) are incomplete elliptic integrals of the first and second kind, respectively, with arguments,

~\theta = \cos^{-1} \biggl(\frac{c}{a} \biggr)

      and      

~k = \biggl[\frac{1 - (b/a)^2}{1 - (c/a)^2} \biggr]^{1/2} \, .

[ EFE, Chapter 3, §17, Eq. (32) ]

See Also


Whitworth's (1981) Isothermal Free-Energy Surface

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